Of course, Franklin Roosevelt meant by this phrase that the United States would support Great Britain in their stand against Nazi Germany by supplying them with weapons and other war materials. The United States had adopted a posture of neutrality in response to events in Europe, but by 1940, Great Britain stood basically alone against Germany, and when Roosevelt gave this speech, Britain had been experiencing months of almost constant aerial assault by the German Luftwaffe. So in effect, the speech was intended to redefine the role of the United States in the conflict. While staying out of the war, the United States would not be neutral, but rather support those nations fighting Germany.
In practice, this led to the Lend-Lease Act, which allowed the loan of war materials and funds to Britain (and later the USSR.) For the American people, it meant several things. The first was that it became clear that the United States would not be able to remain isolated from events in Europe, and that American national security depended on British success in the conflict. As Roosevelt said in his "arsenal of democracy" speech to the American people:
Some of us like to believe that even if Great Britain falls, we are still safe, because of the broad expanse of the Atlantic and of the Pacific...[But] The British people and their allies today are conducting an active war against this unholy alliance. Our own future security is greatly dependent on the outcome of that fight. Our ability to "keep out of war" is going to be affected by that outcome.
Another effect was that American society began to move toward a war footing. The draft had already been instituted when FDR gave the speech, and the American economy began to shift to war production. This contributed to still another effect, which was the rapid expansion of the American economy, already pulling out of the Great Depression. American factories began hiring in record numbers to meet the demand for war materials for the Allies as well as the United States military machine itself. By 1941, unemployment was virtually nonexistent in the American economy.